Simple homogeneous models

Geometrical stability theory is a powerful set of model-theoretic tools that can lead to structural results on models of a simple first-order theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theory (a saturated model) in which the Stone space topology on ultrafilters of definable relations is compact. Here we operate in the more general setting of homogeneous models, which typically have noncompact Stone topologies. A structure M M equipped with a class of finitary relations R R is strongly λ λ -homogeneous if orbits under automorphisms of ( M , R ) (M,R) have finite character in the following sense: Given α α an ordinal > λ ≤ | M | >λ ≤ |M| and sequences a ¯ = { a i : i > α } ā=\{\,a_i:\:i>α \,\} , b ¯ = { b i : i > α } b̄=\{\,b_i:\:i>α \,\} from M M , if